Question

The number of common roots among the 12^th and 30^th roots of unity is ?

• A. $12$
• B. $9$
• C. $8$
• D. $6$

Hint:

For $n$th and $m$th roots of unity, the common solutions are exactly the $\gcd(n,m)$th roots of unity.

Answer 1

The Correct Option is D

Step 1: Roots of unity definition.

The 12^th roots of unity are the complex solutions to $z^{12} = 1$.

The 30^th roots of unity are the complex solutions to $z^{30} = 1$.

Step 2: Common solutions.

A complex number is a common root of both equations if and only if it satisfies

$$z^{12} = 1 \quad\text{and}\quad z^{30} = 1.$$

Equivalently, $z$ is a root of unity whose order divides both 12 and 30.

Step 3: Use greatest common divisor.

The common roots are exactly the $\gcd(12,\,30) = 6$-th roots of unity. Hence there are 6 common roots.